I was listening to music on Youtube pretty much all the time and that amounted to about 35GB of traffic per month in average.
That's 420GB per year and with 7kWh/GB and emissions of approximately 500g/kWh that's a surprising total of roughly 1.5t of CO_2 emissions per year. And a downside is almost non-existant.
Edit: my playlist consists of a small number of songs, so I actually get savings. That would not work when listening to a list that does not actually repeat of course.